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A353004 Numbers k such that 2*k^2 + 29 is semiprime.

%I A353004 #26 Jul 29 2025 14:25:08
%S A353004 29,30,32,35,39,44,50,57,58,61,63,65,72,74,76,84,87,88,89,91,92,94,95,
%T A353004 97,99,102,107,109,113,116,118,120,122,123,125,126,127,134,138,144,
%U A353004 145,146,147,148,149,150,153,154,156,157,163,164,165,166,169,174,175,179,180,182,183,191,194,196,200
%N A353004 Numbers k such that 2*k^2 + 29 is semiprime.
%C A353004 The least positive k for which 2*k^2 + 29 is neither prime nor semiprime is k = 185, which gives 2*k^2 + 29 = 68479 = 31*47^2.
%C A353004 Includes 29*k if 58*k^2 + 1 is prime; Bunyakovsky's conjecture implies there are infinitely many of these. - _Robert Israel_, Jul 29 2025
%H A353004 Robert Israel, <a href="/A353004/b353004.txt">Table of n, a(n) for n = 1..10000</a>
%e A353004 a(5) = 39; 2*39^2 + 29 = 3071 = 37*83 is semiprime.
%p A353004 filter:= proc(k) numtheory:-bigomega(2*k^2+29) = 2 end proc;
%p A353004 select(filter, [$1..1000]); # _Robert Israel_, Jul 29 2025
%t A353004 Select[Range[200], PrimeOmega[2*#^2 + 29] == 2 &] (* _Amiram Eldar_, Apr 15 2022 *)
%o A353004 (Python)
%o A353004 from sympy import primeomega
%o A353004 def semiprime(n): return primeomega(n) == 2
%o A353004 print([k for k in range(140) if semiprime(2*k**2+29)]) # _Michael S. Branicky_, Apr 15 2022
%o A353004 (PARI) isok(k) = bigomega(2*k^2+29) == 2; \\ _Michel Marcus_, Apr 15 2022
%Y A353004 Subsequence of A007642, whose first term not in this sequence is 185.
%Y A353004 Cf. A241554, A352949, A353388.
%K A353004 nonn
%O A353004 1,1
%A A353004 _Rémi Guillaume_, Apr 15 2022